Solid mechanics is concerned with the stress, deformation, and failure of solid materials and structures, and is useful for many end use applications. By way of example, engineering scientists in the fields of mechanical, structural, materials, civil, and aerospace engineering study solid mechanics. As a more particular example, the behavior of soil and rock under loads and stresses may be studied for purposes that may include engineering projects such as tunnels, open-cut excavations, foundation settlements, slope deformations, and embankment response.
Models are available for representing solid mechanics relations in a mathematical format. As a particular example, material constitutive relations and models are mathematical representations of the mechanical response of material relating stress to strain states as well as other aspects of material behavior including time dependent behavior such as heat flow, and creep. Constitutive models represent the material behavior at the point or “element” level and relate a second order symmetric stress tensor with six stress components to a second order strain tensor. One of the earliest and simplest constitutive models is Hook's Law, a mathematical relation that describes linear response of a spring being proportional to an applied force. Plasticity accounts for inelastic behavior by a yield criterion such as the Tresca and von Mises yield criteria to represent material failure. A flow rule determines material response at the yield surface. While these constitutive models are useful to establish a general framework for understanding material behavior, they may be inadequate for fully addressing complex behavior such as non-linearity, hardening and softening, anisotropy, and strain rate dependence.
Advanced models were therefore developed with versatile yield surfaces that expand to capture the maximum past stress experienced by the material and undergo rotation to simulate anisotropy. The Cam-Clay and the modified Cam-Clay models (“MCC”) represent one of the earliest models widely used in geotechnical engineering to represent clay behavior and include an elliptical yield surface. Models were later developed with multiple yield surfaces and bounding surface plasticity. Other models can simulate the response of the soil under seismic condition. In these models soil is viewed as an assemblage of particles and can be used to capture localization.
In modeling a system to study solid mechanics, it may be helpful to consider more than one of the many models that are available. Indeed, different models may be useful for studying different relations, phenomenon, or the like of a particular system. Repeated modeling of the system using different models, however, can be a laborious task. It may require, for instance, locating a reference describing each model, coding a computer to apply the model, and studying the output of the model.
In addition to these problems associated with using multiple models to study a system, many solid mechanics models, with constitutive models as an example, are computationally intensive models that traditionally have been difficult to apply and understand even on an individual basis. Advances in computer hardware technology, however, have given a large cross-section of the engineering community access to inexpensive fast computers that substantially reduce the computational intensity of using the models. Users, some with limited background in numerical analysis techniques, can easily perform in a short period of time what used to be sophisticated lengthy analyses. The users of the complex models, however, need to have a fundamental understanding of the model and its influence on the engineering analysis.
Currently, it is difficult for new users of a complex solid mechanics model such as a constitutive model to develop an understanding of the model without spending significant time and effort in understanding the mathematical formulation, programming the equations, and then performing many simulations to examine the model performance.
Further, users of these complex models face difficulty in understanding and appreciating the model output. For example, constitutive relations are often described in terms of volumes and surfaces and represent quantities in a multidimensional space (e.g., six stress and strain components). The models may be displayed in a two-dimensional space (e.g., stress-strain plots), and may also be represented as a schematic of the volumes representing yield surface, plastic potential and failure surface. Users of constitutive models spend significant time and effort in developing a mental framework for understanding a given constitutive model, its output, and its implication on material behavior.
Unresolved problems in the art therefore remain.